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Question

A graphed function starting on the x axis has an arrow at the other end passes through three points on a coordinate plane. The x-axis ranges from negative 9 to 9 in unit increments and the y-axis ranges from negative 11 to 11 in unit increments.

Which table shows that y=−f(x)
results in a reflection over the x-axis?

(1 point)
Responses

x −f(x)
−2
0
−3
1
−6
2
x f left parenthesis x right parenthesis negative 2 0 negative 3 1 negative 6 2

x −f(x)
2 0
3 1
6 2
x f left parenthesis x right parenthesis 2 0 3 1 6 2

x −f(x)
2 0
3 −1
6 −2
x f left parenthesis x right parenthesis 2 0 3 negative 1 6 negative 2

x −f(x)
−2
0
−3
−1
−6
−2
x f left parenthesis x right parenthesis negative 2 0 negative 3 negative 1 negative 6 negative 2
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Answer 1

To determine which table correctly represents \( y = -f(x) \), we must understand that reflecting a function over the x-axis effectively means that the y-values of the function become their opposites. In other words, if a point on the original function is \((x, f(x))\), then under the transformation \(y = -f(x)\), that point would move to \((x, -f(x))\).

Let's examine each response:

1. Response 1:

   • \( x \) | \( -f(x) \)
   • -2 | ?
   • 0 | ?
   • -3 | ?
   • 1 | ?
   • -6 | ?
   • 2 | ?

   This response does not provide specific values for \( -f(x) \), making it impossible to verify the reflection.

2. Response 2:

   • \( x \) | \( -f(x) \)
   • 2 | 0
   • 3 | 1
   • 6 | 2

   If this is the table for \( -f(x) \), the corresponding \( f(x) \) values need to be checked. The correct opposite values would be:

   • For \( 2 \): \( f(2) = 0 \)
   • For \( 3 \): \( f(3) = -1 \)
   • For \( 6 \): \( f(6) = -2 \) Hence, this arrangement does not represent a reflection.

3. Response 3:

   • \( x \) | \( -f(x) \)
   • 2 | 0
   • 3 | -1
   • 6 | -2

   If \( -f(x) \) has the values given, then:

   • For \( 2 \): \( f(2)\) = 0
   • For \( 3 \): \( f(3) = 1\)
   • For \( 6 \): \( f(6) = 2\) This aligns correctly with the original values under the reflection rule.

4. Response 4:

   • \( x \) | \( -f(x) \)
   • -2 | ?
   • 0 | ?
   • -3 | ?
   • -1 | ?
   • -6 | ?
   • -2 | ?

   Again, not enough information is provided to determine if this is correct.

After analyzing all the responses, Response 3 demonstrates the correct values for \( -f(x) \) after reflecting over the x-axis, since the output values for \( -f(x) \) are exactly the negatives of the input values of \( f(x) \).

Thus, the answer is: Response 3.